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Math for Liberal Studies. Section 3.8: More Modular Arithmetic and Public-Key Cryptography. Modular Multiplication. The modular arithmetic ciphers we have seen use addition and subtraction For example, the Caesar cipher rule is to add 3 to every letter in the message

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modular multiplication
Modular Multiplication
  • The modular arithmetic ciphers we have seen use addition and subtraction
  • For example, the Caesar cipher rule is to add 3 to every letter in the message
  • What about multiplication?
modular multiplication1
Modular Multiplication
  • In an affine cipher, we multiply by a constant number and then add a (possibly different) number
  • In a multiplicative cipher, we just multiply
modular multiplication2
Modular Multiplication
  • For example, let’s encode the message “Attack at daybreak” using an affine cipher where we multiply by 3 and then add 5
modular multiplication3
Modular Multiplication
  • For example, let’s encode the message “Attack at daybreak” using an affine cipher where we multiply by 3 and then add 5
modular multiplication4
Modular Multiplication
  • To encode T, for example, we convert T to 19, multiply by 3 (to get 57), then add 5 (to get 62). Finally we find the remainder, 10
modular multiplication5
Modular Multiplication
  • Now we know how to encode a message using an affine cipher
  • How do we decode?
  • Since the 2nd step is to add 5, we should subtract 5 first
modular multiplication6
Modular Multiplication
  • How do we undo the “multiply by 3” step?
  • The natural answer would be to divide by 3, but that’s not really possible in modular arithmetic
modular multiplication7
Modular Multiplication
  • There is a way to undo the “multiply by 3” step
  • It turns out that what weneed to do is to multiplyby 9
modular multiplication8
Modular Multiplication
  • There is a way to undo the “multiply by 3” step
  • It turns out that what weneed to do is to multiplyby 9
modular multiplication9
Modular Multiplication
  • How does multiplying twice get us back to where we started?
  • Notice that we’re multiplying by 3 and then by 9
  • So we’re multiplying by 3 x 9 = 27 = 1
  • Multiplying by 1 doesn’t change anything
modular multiplication10
Modular Multiplication
  • We say that 3 and 9 are multiplicative inverses modulo 26
  • Also, multiplying by 3 and multiplying by 9 are inverse operations
  • Finding these inverse operations is key to using multiplication to encode and decode messages
searching for inverses
Searching for Inverses
  • Can any multiplication operation be reversed?
  • How about multiplicationby 6?
searching for inverses1
Searching for Inverses
  • Can any multiplication operation be reversed?
  • How about multiplicationby 6?
  • As we can see, both B andO get encoded as G
searching for inverses2
Searching for Inverses
  • If we wanted to decodea message that has a Gin it, there would be noway of knowing if it shouldbe a B or an O
  • We can’t use this operationto encode messages becauseit is not reversible
searching for inverses3
Searching for Inverses
  • The problem is even worseif we try to multiply by 13
  • Every letter gets encodedas A or N
  • It would be impossible to decode a message usingthis cipher
searching for inverses4
Searching for Inverses
  • How can we know in advance whether a particular multiplication operation will be reversible?
  • Whenever we multiply by a number that shares a common factor greater than 1 with the modulus, that operation will not be reversible
searching for inverses5
Searching for Inverses
  • 6 and 26 are both divisible by 2
  • 13 and 26 are both divisible by 13
  • 3 and 26 do not share any common factors other than 1
prime numbers
Prime Numbers
  • For this reason, we will often change the modulus from 26 to a prime number
  • A prime number only has divisors of 1 and itself
  • For example, 27 is not prime (its divisors are 1, 3, 9, and 27), but 29 is prime
prime numbers1
Prime Numbers
  • The advantage of using a prime number modulus is that we no longer have to worry about the numbers we multiply by sharing common factors with the modulus
  • The disadvantage is that our modulus is now greater than 26
prime numbers2
Prime Numbers
  • Suppose we encode the message “Retreat at once” using multiplication by 6, modulo 29
prime numbers3
Prime Numbers
  • Suppose we encode the message “Retreat at once” using multiplication by 6, modulo 29
prime numbers4
Prime Numbers
  • Suppose we encode the message “Retreat at once” using multiplication by 6, modulo 29
  • Notice that we get results like 26 and 27, which we cannot turn back into letters
prime numbers5
Prime Numbers
  • When we use multiplicative and affine ciphers with a modulus greater than 26, we will simply leave our encoded message as a sequence of numbers
prime numbers6
Prime Numbers
  • We know that using prime numbers means that we will always have a reversible multiplicative cipher
  • However, in general, finding the inverse operation can be quite difficult
public key cryptography
Public-Key Cryptography
  • Affine and multiplicative ciphers are examples of asymmetric ciphers
  • This means that the method for decoding messages is not easily determined even if you know the encoding method
  • Asymmetric ciphers allow for a system called public-key cryptography
public key cryptography1
Public-Key Cryptography
  • In a public-key system, everyone has two keys: a method for encoding messages, and a method for decoding messages
  • With a multiplicative cipher, the keys are numbers:
    • The encoding key is the number you must multiply by to encode the message
    • The decoding key is the number used to decode the message
public key cryptography2
Public-Key Cryptography
  • The difference in a public-key system is that the encoding keys are made public
  • Everyone knows everyone else’s encoding key
  • The decoding keys are kept private
  • For a system like this to work, it must be very difficult to determine the decoding keys from the encoding keys
public key cryptography3
Public-Key Cryptography
  • How does a system like this work?
  • Suppose Adam wants to send a message to Beth
  • Adam encodes his messageusing Beth’s public key
public key cryptography4
Public-Key Cryptography
  • Remember that only Beth knows her private decoding key
  • So only Beth will be able to decode the message
  • But anyone is able to senda private message like thisto Beth
public key cryptography5
Public-Key Cryptography
  • Think about it this way: Using Beth’s public key to encode a message is like putting in a special kind of box that only Beth can open
  • In this case suppose that Beth’sboxes are blue, and only Bethcan open blue boxes
  • But anyone can buyblue boxes,and use them to send messagesto her
public key cryptography6
Public-Key Cryptography
  • Suppose Adam’s boxes are red
  • Again, anyone can buy red Adam boxes, but only Adam can open them
  • If Christine sends Adam amessage encoded using hispublic key, she doesn’t haveto worry about Beth intercepting the message anddecoding it
beyond packages signatures
Beyond Packages: Signatures
  • This analogy breaks down a bit, as there is something else we can do with this system
  • Suppose Christine works at a bank and receives a request to transfer a large amount out of Beth’s bank account
  • She wants Beth to send a secure message verifying the transfer
beyond packages signatures1
Beyond Packages: Signatures
  • The problem is that anyone can send a message to Christine posing as Beth
  • Even if they use Christine’s public key to encode the message, this doesn’tensure that the message camefrom Beth, it just makes surethat the message goes to Christine
beyond packages signatures2
Beyond Packages: Signatures
  • Instead, Beth can send a message (such as “I authorize the transfer”) to Christine encoded with Beth’s private key
  • Now anyone can decode this message (using Beth’s public key), but only Beth could have encoded it, since only Beth knows that private key
  • In this way, it is guaranteed that the message came from Beth
real world cryptography
Real-World Cryptography
  • Notice that affine and multiplicative ciphers are substitution ciphers
  • This means that they are susceptible to techniques like frequency analysis
  • These methods are unsuitable to use for public-key cryptography, which requires that the codes be virtually unbreakable
real world cryptography1
Real-World Cryptography
  • A common system in use today is RSA cryptography, which relies on the same kinds of ideas we have studied (prime numbers and inverse operations in modular arithmetic)
  • However, the prime numbers we use for RSA are many digits long
real world cryptography2
Real-World Cryptography
  • For example, here is a typical security message used by a web browser
  • “128 bit” means that the encryption key could be as large as 2128, which a 39-digit number
real world cryptography3
Real-World Cryptography
  • Even with a computer checking a billion keys per second, it would take up to 10 thousand billion billion years to check them all
  • This is far, far longer than the age of the universe
  • RSA relies on the size of the key being large enough to make breaking the code take an impossibly long amount of time